Connection with GHC's Constraint Solver

The solver for the type nats is implemented as an
extra stage in GHC's constrraint solver (see TcInteract.thePipeline).

Generating Evidence

The solver produces evidence (i.e., proofs) when computing new "given"
constraints, or when solving existing "wanted" constraints.
The evidence is constructed by applications of a set of pre-defined
rules. The rules are values of type TypeRep.CoAxiomRule.
Conceptually, rules have the form:

name :: forall tyvars. assumptions => conclusion

The rules have the usual logical meaning: the variables are universally
quantified, and the assumptions imply the concluson. As a concrete example,
consider the rule for left-cancellation of addtion:

AddCanceL :: forall a b c d. (a + b ~ d, a + c ~ d) => b ~ c

The type CoAxiomRule also supports infinte literal-indexed families
of simple axioms using constructor CoAxiomTyLit. These have the form:

name(l_1 .. l_n) :: conclusion

In this case conclusion is an equation that contains no type variables
but may depend on the literals in the name of the family. For example,
the basic definitional axiom for addition, TcTypeNatsRules.axAddDef,
uses this mechanism:

AddDef(2,3) :: 2 + 3 ~ 5

At present, the assumptions and conclusion of all rules are equations between
types but this restriction is not important and could be lifted in the future.

The rules used by the solver are in module TcTypeNatsRules.

The Solver

The entry point to the solver is TcTypeNats.typeNatStage.

We start by examining the constraint to see if it is obviously unsolvable
(using function impossible), and if so we stash it in the
constraint-solver's state and stop. Note that there is no assumption that
impossible is complete, but it is important that it is sound, so
if impossible returns True, then the constraint is definitely unsolvable,
but if impossible returns False, then we don't know if the constraint
is solvable or not.

The rest of the stage proceeds depending on the type of constraint,
as follows.

Given Constraints

Given constraints correspond to adding new assumptions that may be used
by the solver. We start by checking if the new constraint is trivial
(using function solve). A constraint is considered to be trivial
if it matches an already existing constraint or a rule that is known
to the solver. Such given constraints are ignored because they do not
contribute new information. If the new given is non-trivial, then it
will be recorded to the inert set as a new fact, and we proceed
to "interact" it with existing givens, in the hope of computing additional
useful facts (function computeNewGivenWork).

IMPORTANT: We assume that "given" constraints are processed before "wanted"
ones. A new given constraint may be used to solve any existing
wanted, so every time we added a new given to the inert set we should
move all potentially solvable "wanted" constraint from the
inert set back to the work queue. We DON'T do this, because it is
quite inefficient: there is no obvious way to compute which "wanted"s
might be affected, so we have to restart all of them!

The heart of the interaction is the function interactCt, which
performs one step of "forward" reasoning. The idea is to compute
new constraints whose proofs are made by an application of a rule
to the new given, and some existing givens. These new constraints are
added as new work, to be processed further on the next iteration of
GHC's constraint solver.

Aside: when we compute the new facts, we check to see if any are
obvious contradictions. This is not strictly necessary because they
would be detected on the next iteration of the solver. However, by doing
the check early we get slightly better error messages because
we can report the original constraint as being unsolvable (it leads
to a contradiction), which tends to be easier to relate to the original
program. Of course, this is not completely fool-proof---it is still
possible that a contradiction is detected at a later iteration.
An alternative idea---not yet implemented---would be to examine the
proof of a contradiction and extract the original constraints that lead
to it in the first place.

Derived Constraints

Derived constraints are facts that are implied by the constraints
in the inert set. They do not have complete proofs because
they may depend on proofs of as yet unsolved wanted constraints.
GHC does not associate any proof terms with derived constraints (to keep things simple?).
In the constraint solver, they are mostly used as "hints". For example,
consider the wanted constraint 5 + 3 ~ x, where x is a
free unification variable. These are the steps we'll take to solve
the constraint:

The type-nat solver processes derived constraints in a similar fashion
to given constraints (computeNewDerivedWork): it checks to see if they are trivially known
and, if not, then it tries to generate some additional derived constraints.
The main difference is that derived constraints can be interacted
with all existing constraints to produce new facts, while given
constraints only interact with other givens.

Wanted Constraints

The main purpose of the solver is to discharge wanted constraints
(the purpose of processing given and derived constraints is to help
solve existing wanted goals). When we encounter a new wanted goals
we proceed as follows:

Try to solve the goal, using a few different strategies:

Try to see if it matches the conclusion of an iff rule (solveIff). Aassumptions of rule become new wanted work.

Try to see if it matches an axiom exactly (solve)

Try the ordering solver for <= goals (solveLeq)

Try to use a (possibly synthesized) assumption

If that didn't work:

Wanted is added to the inert set

Check to see if any of the existing wanteds in the inert set can be solved in terms of the new goal (reExamineWanteds)